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In this work, we extend the recent methods Kapl et al. Comput. Aided Geom. Des.<jats:bold>52\u201353<\/jats:bold>:75\u201389, 2017, Kapl et al. Comput. Aided Geom. Des.<jats:bold>69<\/jats:bold>:55\u201375, 2019 and Kapl and Vitrih J. Comput. Appl. Math.<jats:bold>335<\/jats:bold>:289\u2013311, 2018, Kapl and Vitrih J. Comput. Appl. Math.<jats:bold>358<\/jats:bold>:385\u2013404, 2019 and Kapl and Vitrih Comput. Methods Appl. Mech. Engrg.<jats:bold>360<\/jats:bold>:112684, 2020 for the construction of<jats:italic>C<\/jats:italic><jats:sup>1<\/jats:sup>-smooth and<jats:italic>C<\/jats:italic><jats:sup>2<\/jats:sup>-smooth isogeometric spline spaces over particular planar multi-patch geometries to the case of<jats:italic>C<\/jats:italic><jats:sup><jats:italic>s<\/jats:italic><\/jats:sup>-smooth isogeometric multi-patch spline spaces of degree<jats:italic>p<\/jats:italic>, inner regularity<jats:italic>r<\/jats:italic>and of a smoothness<jats:italic>s<\/jats:italic>\u2265\u20091, with<jats:italic>p<\/jats:italic>\u2265\u20092<jats:italic>s<\/jats:italic>+\u20091 and<jats:italic>s<\/jats:italic>\u2264<jats:italic>r<\/jats:italic>\u2264<jats:italic>p<\/jats:italic>\u2212<jats:italic>s<\/jats:italic>\u2212\u20091. More precisely, we study for<jats:italic>s<\/jats:italic>\u2265\u20091 the space of<jats:italic>C<\/jats:italic><jats:sup><jats:italic>s<\/jats:italic><\/jats:sup>-smooth isogeometric spline functions defined on planar, bilinearly parameterized multi-patch domains, and generate a particular<jats:italic>C<\/jats:italic><jats:sup><jats:italic>s<\/jats:italic><\/jats:sup>-smooth subspace of the entire<jats:italic>C<\/jats:italic><jats:sup><jats:italic>s<\/jats:italic><\/jats:sup>-smooth isogeometric multi-patch spline space. We further present the construction of a basis for this<jats:italic>C<\/jats:italic><jats:sup><jats:italic>s<\/jats:italic><\/jats:sup>-smooth subspace, which consists of simple and locally supported functions. Moreover, we use the<jats:italic>C<\/jats:italic><jats:sup><jats:italic>s<\/jats:italic><\/jats:sup>-smooth spline functions to perform<jats:italic>L<\/jats:italic><jats:sup>2<\/jats:sup>approximation on bilinearly parameterized multi-patch domains, where the obtained numerical results indicate an optimal approximation power of the constructed<jats:italic>C<\/jats:italic><jats:sup><jats:italic>s<\/jats:italic><\/jats:sup>-smooth subspace.<\/jats:p>","DOI":"10.1007\/s10444-021-09868-5","type":"journal-article","created":{"date-parts":[[2021,5,30]],"date-time":"2021-05-30T14:02:20Z","timestamp":1622383340000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Cs-smooth isogeometric spline spaces over planar bilinear multi-patch parameterizations"],"prefix":"10.1007","volume":"47","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8153-5987","authenticated-orcid":false,"given":"Mario","family":"Kapl","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Vito","family":"Vitrih","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2021,5,30]]},"reference":[{"key":"9868_CR1","doi-asserted-by":"publisher","first-page":"1073","DOI":"10.1016\/j.cma.2014.11.038","volume":"284","author":"C Anitescu","year":"2015","unstructured":"Anitescu, C., Jia, Y., Zhang, Y.J., Rabczuk, T.: An isogeometric collocation method using superconvergent points. 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